Questions tagged [algebraic-surfaces]
An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
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Enriques-Kodaira classification of minimal resolution of surface with quotient singularities
Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient ...
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Embedding degree 1 Del Pezzo surfaces in $\mathbb{P}(1,1,2,3)$
In the projective bundle $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\oplus \mathcal{O})\rightarrow\mathbb{P}^1$ consider the hypersruface
$$
X := \{a_{00}y_0^2+a_{01}y_0y_1+a_{02}y_0y_2+a_{11}...
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Contraction of $(-1)$ curve and extremal ray
I want to prove Castelnuovo's contraction theorem by Mori's contraction theorem.
Question. How can one show that a $(-1)$ curve on a smooth surface is an extremal ray?
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Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
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Reference request The support of $f$-nef divisor
I'm seaching for a proof of the theorem below.
Do you know any reference?
Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
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Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
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One question about Manetti surface
I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof.
Theorem 5.2 states that fixed a ...
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Question about surface singularities
Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities,
I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
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Equality case of the log-Bogomolov-Miyaoka-Yau inequality
The Bogomolov-Miyaoka-Yau inequality for sufaces says that if $X$ is a smooth projective minimal surface of general type then $c_1(X)^2 \le 3 c_2(X)$. It is a theorem of Yau (I think) that equality ...
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Formula for Pushforward of structure sheaf for branched coverings
I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces.
Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $...
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On the positivity of cotangent bundle of elliptic surfaces
I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want.
Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon ...
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Action on Enriques surface by sections of Jacobian fibration
A question about a statement in Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $\pi: Y \to \mathbb{P}^1$ be a special elliptic pencil of complex Enriques ...
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What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
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(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve
Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
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Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface
Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...
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